Let $[\mathcal{A},\alpha]$ and $[\mathcal{B},\beta]$ be Banach operator ideals between Banach spaces. ($\alpha$ and $\beta$ are the respective ideal norms)
Suppose that $[\mathcal{A}(E,F),\alpha]=[\mathcal{B}(E,F),\beta]$, for every pair of Banach spaces $E, F$ over $\mathbb{R}$, ¿is it true that $[\mathcal{A},\alpha]=[\mathcal{B},\beta]$ in general?
I suspect that the answer is negative. I was thinking about properties or results that hold over real Banach spaces, but not over complex Banach spaces, like the Mazur-Ulam Theorem, but maybe it is not possible to capture that property in terms of ideals.